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Topic 17 of 22

Non-Abelian Gauge Theory

QED is an Abelian gauge theory — U(1) generators commute. The weak and strong forces require non-Abelian gauge theories based on SU(2) and SU(3), whose generators do not commute: [T^a, T^b] = if^{abc}T^c. This non-commutativity has a dramatic physical consequence: the gauge bosons (gluons, W, Z) carry charge and interact with each other, producing cubic and quartic self-interaction vertices absent from electromagnetism — and ultimately responsible for asymptotic freedom.

Key Concepts

  • Lie algebra [T^a, T^b] = if^{abc}T^c with structure constants f^{abc}; SU(N) has N²−1 generators
  • Non-Abelian gauge field: one A^a_μ per generator; F^a_μν = ∂μA^a_ν − ∂νA^a_μ + gf^{abc}A^b_μA^c_ν
  • Yang-Mills Lagrangian: ℒ_YM = −¼F^a_μν F^{aμν} — generates 3-boson and 4-boson vertices
  • Covariant derivative: Dμ = ∂μ − igT^aA^a_μ (representation-dependent)
  • SU(3): 8 generators (Gell-Mann matrices), f^{abc} = ε^{abc} for SU(2) subset
  • Color factors: C_F = (N²_c−1)/2N_c = 4/3, C_A = N_c = 3 (quark-gluon and gluon-gluon couplings)

Key Equations

SU(N) Lie algebra
[Ta,Tb]=ifabcTc,tr[TaTb]=TFδab,  TF=12[T^a,T^b]=if^{abc}T^c,\quad\mathrm{tr}[T^aT^b]=T_F\delta^{ab},\;T_F=\tfrac{1}{2}
Non-Abelian field strength
Fμνa=∂μAνa−∂νAμa+gfabcAμbAνcF^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+gf^{abc}A^b_\mu A^c_\nu
Yang-Mills Lagrangian
LYM=−14FμνaFaμν\mathcal{L}_{\rm YM}=-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}
Casimir operators
CF=Nc2−12Nc=43,CA=Nc=3(for SU(3))C_F=\frac{N_c^2-1}{2N_c}=\frac{4}{3},\quad C_A=N_c=3\quad(\text{for SU(3)})
Worked Example

Example Problem

Problem

SU(2) has structure constants f^{abc} = ε^{abc}. Write the 3-gluon coupling vertex factor from the F^a_μν F^{aμν} Lagrangian.

Solution

Expanding F^a_μν F^{aμν}, the cross term 2(∂A)(gfAAA) gives a vertex with three gauge boson lines. The vertex factor is gf^{abc}[gμν(k₁−k₂)λ + gνλ(k₂−k₃)μ + gλμ(k₃−k₁)ν] where k₁,k₂,k₃ are the incoming momenta — the characteristic momentum-dependent 3-boson vertex of Yang-Mills theory.

Key Takeaways

  • Non-Abelian structure constants f^{abc} ≠ 0 mean gauge bosons carry charge and interact with each other — unlike photons in QED
  • The Yang-Mills Lagrangian −¼F^a_μν F^{aμν} produces 3-boson and 4-boson self-interaction vertices when expanded in A
  • Gluon self-interactions contribute negatively to the QCD beta function (anti-screening) — the origin of asymptotic freedom, impossible in QED
  • Color algebra (Casimir operators C_F and C_A) determines the relative strength of quark-gluon vs. gluon-gluon interactions
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